Information Geometry

Dually flat manifolds

Let \((M,g,\nabla,\nabla^*)\) be a dually connected Riemannian manifold. It will be called a dually flat manifold if \(\nabla\) is flat (and consequentially \(\nabla^*\) too).

If \(M\) is dually flat, then there exists locally two affine coordinate systems \(\theta=(\theta^1,\dots,\theta^n)\) and \(\eta=(\eta_1,\dots,\eta_n)\) for \(\nabla\) and \(\nabla^*\) respectively, with local basis \(\partial_i \coloneqq \frac{\partial}{\partial \theta^i}\) and \(\partial^j \coloneqq \frac{\partial}{\partial \eta_j}\), which are affine in the sense that \(\nabla_{\partial_i} \partial_j \equiv 0\) and \(\nabla^*_{\partial^i} \partial^j \equiv 0\). In other words, \(\theta\) vanishes the Christoffel symbols of \(\nabla\) and \(\eta\) vanishes the Christoffel symbols of \(\nabla^*\).

In the case where \(\nabla\) and \(\nabla^*\) are duals, we can construct such coordinate systems \(\theta\) and \(\eta\) with the additional property that their local frames are duals:

\[\langle \partial_i, \partial^j \rangle = \langle \partial^i, \partial_j \rangle = \delta_{ij}.\]

Such coordinate systems are called dual coordinate systems for \((M,g,\nabla,\nabla^*)\).